So far, we have considered general dynamical variables represented by
general linear
operators acting in ket space. However, in classical mechanics, the most important
dynamical variables are those involving
position and momentum. Let us investigate
the role of such variables in quantum mechanics.

In classical mechanics, the position
and momentum
of some component of a dynamical system are represented as real numbers which,
by definition, commute.
In quantum mechanics, these quantities are represented
as non-commuting linear Hermitian operators acting in a ket space
that represents all of the possible states of the system. Our first task is
to discover a quantum mechanical replacement for the classical result
.